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Properties

Label 12.0.27482671235...9856.1
Degree $12$
Signature $[0, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 7^{10}\cdot 19^{10}$
Root discriminant $611.77$
Ramified primes $2, 3, 7, 19$
Class number $11804184$ (GRH)
Class group $[6, 6, 327894]$ (GRH)
Galois group $C_6\times C_2$ (as 12T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1432809, 0, 743627871, 0, 174325095, 0, 11626860, 0, 166383, 0, 798, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^12 + 798*x^10 + 166383*x^8 + 11626860*x^6 + 174325095*x^4 + 743627871*x^2 + 1432809)
 
gp: K = bnfinit(x^12 + 798*x^10 + 166383*x^8 + 11626860*x^6 + 174325095*x^4 + 743627871*x^2 + 1432809, 1)
 

Normalized defining polynomial

\( x^{12} + 798 x^{10} + 166383 x^{8} + 11626860 x^{6} + 174325095 x^{4} + 743627871 x^{2} + 1432809 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $12$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2748267123526559548681146214649856=2^{12}\cdot 3^{18}\cdot 7^{10}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $611.77$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(3457,·)$, $\chi_{4788}(3299,·)$, $\chi_{4788}(2015,·)$, $\chi_{4788}(1,·)$, $\chi_{4788}(4777,·)$, $\chi_{4788}(4103,·)$, $\chi_{4788}(1741,·)$, $\chi_{4788}(1775,·)$, $\chi_{4788}(277,·)$, $\chi_{4788}(121,·)$, $\chi_{4788}(2747,·)$, $\chi_{4788}(4415,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{399} a^{6}$, $\frac{1}{399} a^{7}$, $\frac{1}{1197} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{1197} a^{9} + \frac{1}{3} a^{3}$, $\frac{1}{2368926497557053} a^{10} - \frac{654849203}{87738018428039} a^{8} - \frac{292870908598}{263214055284117} a^{6} + \frac{1201811441056}{5937159141747} a^{4} - \frac{822467791583}{1979053047249} a^{2} - \frac{29370137759}{659684349083}$, $\frac{1}{945201672525264147} a^{11} - \frac{8095017200}{124680341976687} a^{9} - \frac{98343095276}{87738018428039} a^{7} + \frac{672100794458467}{2368926497557053} a^{5} - \frac{2674681560889}{5937159141747} a^{3} - \frac{134141711000}{1979053047249} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{6}\times C_{6}\times C_{327894}$, which has order $11804184$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $5$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 150874.89368738522 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_6$ (as 12T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 12
The 12 conjugacy class representatives for $C_6\times C_2$
Character table for $C_6\times C_2$

Intermediate fields

\(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-57}) \), \(\Q(\sqrt{133}) \), 3.3.1432809.1, \(\Q(\sqrt{-21}, \sqrt{-57})\), 6.0.2759153551366464.4, 6.0.7489131067994688.3, 6.6.273041236853973.3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ R ${\href{/LocalNumberField/23.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
$7$7.12.10.2$x^{12} + 35 x^{6} + 441$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$19$19.6.5.3$x^{6} - 4864$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.3$x^{6} - 4864$$6$$1$$5$$C_6$$[\ ]_{6}$